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---

:t lw,[

pudV

=

Saw, + f pbdV, lw,

(BM3)

8

1 The Equations of Motion

moving fluid

/

FIGURE

1.1.4. Wt is the image of Was particles of fluid in W ftow for timet.

that is, the rate of change of momentum of a moving piece of fluid equals the total force (surface stressesplus body forces) acting on it. These two forms of balance of momentum (BMl) and (BM3) are equivalent. To prove this, we use the change of variables theorem to write dd

t

f pudV = dd f (pu)( 0. This agrees with the common experience that increasing the surrounding pressure on a volume of fluid causes a decrease in occupied volume and hence an increase in density. Gases can often be viewed as isentropic, with p=Ap'~',

where A and 'Y are constants and 'Y ::::: 1. Here,

w =

J

P "(As'~'- 1

s

"fAp'~'- 1

ds = --'--"(-1

and

Ap'~'-l

E=---. "(-1

Cases 1 and 2 above are rather opposite. For instance, if p = p0 is a constant for an incompressible fluid, then clearly p cannot be an invertible function of p. However, the case p = constant may be regarded as a limiting case p'(p) ----> oo. In case 2, p is an explicit function of p (and therefore 3 0ne can carry this even further and use balance of energy and its invariance under Euclidean motions to derive balance of momentum and mass, a result of Green and Naghdi. See Marsden and Hughes [1994] for a proof and extensions of the result that include formulas such as p = p 2 8c:jßp amongst the consequences as weil.

16

1 The Equations of Motion

depends on u through the coupling of p and u in the equation of continuity); in case 1, p is implicitly determined by the condition div u = 0. We shall discuss these points again later. Finally, notice that in neither case 1 or 2 is the possibility of a loss of kinetic energy due to friction taken into account. This will be discussed at length in §1.3. Given a fluid flow with velocity field u(x, t), a streamline at a fixed time is an integral curve of u; that is, if x(s) is a streamline at theinstaut t, it is a curve parametrized by a variable, say s, that satisfies

dx

ds = u(x(s), t),

t fixed.

We define a fixed trajectory to be the curve traced out by a particle as time progresses, as explained at the beginning of this section. Thus, a trajectory is a solution of the differential equation

dx dt

= u(x(t), t)

with suitable initial conditions. If u is independent of t (i.e., OtU = 0), streamlines and trajectories coincide. In this case, the flow is called sta-

tionary.

Bernoulli's Theorem In stationary isentropic ftows and in the absence of external forces, the quantity

is constant along streamlines. The same holds for homogeneaus (p = constant in space = Po) incompressible ftow with w replaced by pj Po· The conclusions remain true if a force b is present and is conservative; i. e., b = -V cp for some function cp, with w replaced by w + cp.

Proof From the table of vector identities at the back of the book, one has !V(IIull 2 ) = (u · V)u + u x (V x u). Because the flow is steady, the equations of motion give

(u·V)u=-Vw and so

V {!llull 2 + w